# Full text of "Mathematical And Physical Papers - Iii"

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```ON THE MOTION OF PENDULUMS.                       49
the last figure being in each case in the 12th place of decimals. We thus get
9r-*r'(i)=- 1-9635102,   A = + '5772158* ............ (106).
34. When m is large, it will be more convenient to employ series according to descending powers of a. Observing that the general term of FB (a) as given by (88), in which Dr = 0, is
wo get for the general term of F^ (a)
M r,       ___   [1 . 3. - .(2. - 3)]=       f    (2t- 1)2 _ 2t-l] *-     ;            2.'4.".(2*-2y(4«a)pio*r  2t.*«*       2a  }'
and the expression within brackets is equivalent to
whence
(             1   %         V2  3   5
dW '((A — fyV"m"       i• j__1 __        '        •      •*••**• «'
and wo find by actual division
85.    When many terms are required, the calculation  of the \                 coefficients may be facilitated in the following manner.
j                       Assuming afr^ (a) = v (<t) Al, (M), we have
*v (") -K' ff oo -«" ^ («) + «"* (woaj/y; w-
Substituting  in   the  difle.re.ntial   (ujuation   (<S5)   which   I<\ lias  to satisfy, we p;et
f/.i/' (a) + [v (a)}'2 - uM = 0...............(107).
Assuming
/ \                  iiit/'    \~1' _i  -i  f    N""* _L         ^ i ^^^
* (A is in Hid the well-known transcendent called Hitler's Otiwt.itnt., tho value of which is -r>77'21;*><>(;•{<) it<«. This, whic.li I oiitfht to lia,v<5 known, was iiointe.d out to me just after the puhiieation of the paper by my friend Prof. V. Newman.]
S.    III.                                                                                                                        4```